# nLab complex analytic space

### Context

#### Complex geometry

Could not include synthetic complex geometry - contents

# Contents

## Idea

The notion of complex analytic space is the notion of analytic space in complex geometry; the generalization of the notion of complex manifold to spaces with singularities.

## Definition

A complex analytic test space is a common vanishing locus of a set of holomorphic functions $\mathbb{C}^n \to \mathbb{C}$. This is naturally a locally ringed space over the complex numbers $\mathbb{C}$. A complex analytic space is a locally ringed space over $\mathbb{C}$ that is locally isomorphic to such a complex analytic test space.

## Properties

### Local contractibility

A smooth complex analytic space is locally isomorphic to a polydisc and hence locally contractible. See also (Berkovich, p.2).

### GAGA theorems

Comparison to complex algebraic varieties (GAGA):

## References

Introductions include

• Brian Osserman, Complex varieties and the analytic topology (pdf)

Generalization of smooth complex analytic spaces to smooth $p$-adic analytic spaces is discussed in

Discussion in higher geometry/higher algebra (derived complex analytic spaces) is in

Last revised on May 22, 2017 at 08:20:41. See the history of this page for a list of all contributions to it.